The Millennium Problems; The Navier-Stokes Equation

In May 2000, the Clay Mathematics Institute stated 7 of the perhaps most significant problems in Mathematics at the time. One of the most notable problems of them all is the Navier-Stokes Equations.

The Navier-Stokes Equations came about from applying Newton’s Second Law to fluid dynamics; they describe the relations between pressure, temperature, velocity and density of any moving fluid. Think of any fluid and these equations will explain their behaviour. They subsequently become quite useful, describing the physics of many phenomena, not necessarily in physics, but in other fields as well; they are often used to model weather, ocean currents, water flow in a river, or the flow of air around the wing of a plane; aiding in the design of aircraft and automobiles, the study of blood flow, and the analysis of pollution. Surprisingly, they have also been quite important in the world of gaming as well, and have also been used to study magneto hydrodynamics (assuming you model the equations with Maxwell’s equations, which can help understand how stars and galaxies form – this means we could also potentially model the growth of the sun as well as other significant stars in our galaxy.

While we do have the equations and while solutions may exist with the equation, they are only behaved in 2 dimensions, yet we live in 3 dimensions (assuming you ignore the 4^th dimension of space-time), which affects the equations quite a lot, and we mathematicians have been unable to figure out why these equations do not work in further dimensions. This may be because there is either no way of understanding these equations in such dimensions, or of the possibility that we have not made enough progress in order to potentially find out the solution for further dimensions – meaning we may not be able to solve this problem until we have reached the stage in advancements where we’d understand the problem further.

This Millennium Problem, like all the others, has a prize of 1 million dollars. Only one of the seven problems has been solved, which was the Poincare Conjecture by Grigori Perelman in 2003. Shockingly, he refused to accept the million dollar prize, which shows how many mathematicians out there are not doing such questions for the sake of money, but to further our understanding of such mathematical matters, which often change the course of history as well. Now, the CMI is not asking you to exactly ‘solve’ the problem, but to “further our understanding of the Navier-Stokes Equations” which suggests that we do not need to solve it – we could just prove that it cannot be solved. And so the question remains; who will further our understanding?